The internal bisector of [tex]\angle A[/tex] of triangle [tex]ABC[/tex] meets side [tex]BC[/tex] at [tex]D[/tex]. A line drawn through [tex]D[/tex] perpendicular to [tex]AD[/tex] intersects the side [tex]AC[/tex] at [tex]E[/tex] and the side [tex]AB[/tex] at [tex]F[/tex]. If [tex]a, b, c[/tex] represent the sides of triangle [tex]\triangle ABC[/tex], then which of the following is/are true?
A) [tex]AE[/tex] is the harmonic mean of [tex]b[/tex] and [tex]c[/tex].
B) [tex]AD=\frac{2bc}{b+c} \cos \frac{A}{2}[/tex]
C) [tex]EF=\frac{4bc}{b+c} \sin \frac{A}{2}[/tex]
D) The triangle [tex]AEF[/tex] is isosceles.​